Dr. Algebra

Quadratic Equations are equations that have a variable term with an exponent of ‘2’ and no other variable terms with higher exponents.  The following are all examples: 

• • [katex]3x^2+4x-6=0[/katex]
  • [katex]y^2-6=0[/katex]
  • [katex]ax^2+bx+c=0[/katex]
  • [katex]9z^2=4z-6[/katex]
  • [katex]g^2-2g=1[/katex]

All quadratic equations have exactly TWO solutions (also called roots, or zeroes).  The solutions will occur in one of several different ways.

1.  Two distinct real root such as: 
   • [katex]x=3, and x=7[/katex]
2. Single real root, (or doubled root)
   
   • [katex]x=4, and x=4[/katex]
3. Two complex roots (conjugates)
   • [katex]x=4+6j, and x=4-6j, and x=4[/katex]
   •  
   • There are many methods to solve quadratic equations.  Some methods are more useful than others but as long as you are able to solve the equation, it usually doesn’t matter which method you use.

• • Completing the Square
  • Graphing
  • Factoring
  • Quadratic Formula

Quadratic equations can have several types of solutions (also known as roots, or zeros):

• • • Two REAL roots such as x=5 and x=-5.
    • A single but repeated root such as x=5 and x=5.
    • A solution of x=0.
    • Two complex roots such as x=3+4j and x=3-4j.
      • Note these always occur in pairs called complex conjugates.

Whichever you intend to convey.  Can you see the difference?